这篇论文说一比特均值估计不用交互也能最优，解决了COLT 2026开放问题，统计方向的可以看看。
arXiv:2608.02538 论文研究一比特均值估计问题，其中每个样本只用一个二进制消息表示。作者构造了完全非自适应的随机协议，在分布均值位于[-λ,λ]、绝对k阶中心矩不超过σ^k的条件下，匹配了此前两阶段自适应协议的最优样本复杂度。在目标精度 ε、置信度 1-δ 下，当 k>2 时样本复杂度为 (σ/ε)^2 log(1/δ)，当 k=2 时为 (σ/ε)^2 log(σ/ε) log(1/δ)，当 1<k<2 时为 (σ/ε)^{k/(k-1)} log(1/δ)。该结果给出了 COLT 2026 开放问题“交互是否必要”的否定答案。
Interaction Is Not Necessary for Order-Optimal 1-Bit Mean Estimation
This paper is concerned with one-bit mean estimation, where each independent sample is represented by a single binary message. We consider distributions on $\mathbb{R}$ with mean in $[-λ,λ]$ and absolute $k$-th central moment at most $σ^k$, where $k>1$ is fixed. For this class, previous work attained the optimal sample complexity for general queries using a two-stage protocol. The first stage localizes the mean. The second-stage queries are chosen after localization and refine the estimate around the decoded center. We show that this interaction can be avoided by constructing a randomized fully non-adaptive protocol that fixes all queries before observing the data and matches the optimal adaptive sample complexity. For target accuracy $ε$ and confidence $1-δ$, its sample complexity scales as \[ \log\fracλσ + \begin{cases} (σ/ε)^2\log(1/δ), & k>2,\\ (σ/ε)^2\log(σ/ε)\log(1/δ), & k=2,\\ (σ/ε)^{k/(k-1)}\log(1/δ), & 1<k<2, \end{cases} \] up to constants depending only on $k$. In the range covered by the known lower bound, this rate is minimax optimal even among fully adaptive protocols. This gives a negative answer to the COLT 2026 open problem asking whether interaction is necessary for order-optimal one-bit mean estimation with general queries \citep[Open Problem~1]{lau2026open}.